Theory  of  Pliasemeters.  83 
concerned  must  necessarily  contain  an  air-gap  to  permit  the 
movement  of  this  coil,  and  for  all  dimensions  suitable  for 
instruments  this  air-gap  must  be  such  that  the  reluctance  of 
the  iron  path  is  negligibly  small  compared  with  that  of  the 
gap.  The  flux  densities  in  the  iron  under  these  circum- 
stances can  only  reach  moderate  values  for  which  the  per- 
meability is  fairly  constant,  while  even  a  considerable  change 
in  the  permeability  will  not  appreciably  affect  the  magnetic 
reluctance  of  the  circuit.  Similar  considerations  show  that 
hysteresis  cannot  produce  any  sensible  effect  in  magnetic 
circuits  the  reluctance  of  which  is  almost  entirely  that  of 
an  air  path. 
We  may  therefore  assume  that,  however  unsymmetrical 
the  windings  of  the  three  fixed  coils  may  be,  the  induction 
density  B  at  any  point  on  the  conductors  *  of  the  moving  coil 
for  any  deflexion  x  follows  the  law 
B  =  A^  +  A2F2  +  A3F3,     .     .     .     .     (1) 
where  Fx,  F2,  F3  are  three  quantities  dependent  only  on  x 
and  quite  independent  of  time,  while  Al7  A2,  and  A3  are  the 
instantaneous  values  of  the  currents  in  the  three  coils. 
If  the  current  in  the  moving  coil  is  denoted  by  V,  the 
product  BV  represents  the  momentary  value  of  the  force 
per  unit  length  acting  on  the  portion  of  the  coil  considered. 
It  will  also  be  seen  that  we  can  regard  the  quantities  F  as  line 
averages  for  the  whole  moving  coil  (for  the  deflexion  x)  and 
that  the  turning  moment  acting  on  the  latter  may  be  written 
.  BV  =  F1^V  +  F2^V  +  F8A3V,      ...     (2) 
where,  if  the  currents  are  constant  AV  denotes  the  product 
of  the  corresponding  quantities  after  taking  account  of  their 
sign,  while  if  the  currents  are  periodic  the  expression  repre- 
sents the  average  value  of  such  product. 
Now  as  the  moving  coil  is  not  provided  with  control,  if 
the  above  quantity  is  not  equal  to  zero  the  coil  will  turn 
until  this  is  the  case.  Such  a  position  must  exist,  since  if 
the  coil  be  turned  through  180  degrees  in  either  direction,  the 
torque  acting  on  the  coil  will  be  reversed  in  sign,  and  thus, 
as  the  coil  turns,  must  pass  through  a  zero  value  in  either 
case.  One  of  these  corresponds  with  a  stable,  and  the  other 
with  an  unstable,  position  of  equilibrium. 
If  the  second  member  of  (2)  be  equated  to  zero,  we  get 
an  equation  for  the  deflexion  x,  the  solution  of  which  must 
be  independent  of  the  absolute,  though  not  of  the  relative, 
*  Strictly  speaking  B  is  the  component  of  the  flux  density  perpen- 
dicular to  both  the  conductor  and  the  direction  of  motion. 
G2 
